Classification theory of topological insulators with Clifford algebras and its application to interacting fermions. Takahiro Morimoto.
|
|
- Dorthy Wood
- 5 years ago
- Views:
Transcription
1 QMath13, 10 th October 2016 Classification theory of topological insulators with Clifford algebras and its application to interacting fermions Takahiro Morimoto UC Berkeley
2 Collaborators Akira Furusaki (RIKEN) Christopher Mudry (PSI) Morimoto, Furusaki, PRB 88, (2013) Morimoto, Furusaki, Mudry, Phys. Rev. B 91, (2015)
3 Plan of this talk Introduction Topological insulators and superconductors Ten fold way classification Classification theory of topological insulators Massive Dirac Hamiltonian Clifford algebras and classifying spaces Application to topological crystalline insulators Breakdown of ten fold way classification with interactions Dynamical mass terms Nonlinear sigma model
4 Topological insulator/superconductor is : A system of non-interacting fermions with a band gap. Band insulator Superconductor with a full gap (BdG equation) Characterized by a non-trivial topological number (Z or Z2). Accompanied with a gapless surface state. IQHE B QSHE Majorana chain
5 Topology of energy band Energy band k E(k) Bloch wavefunction k u(k) > Insulator E Brillouin zone Valence wavefunctions k Valence band: u(k) k u(k) > Non trivial way that the Brillouin zone wraps the space of valence wavefucntion. = Topological insulators and superconductors
6 Integer Quantum Hall Effect B xy H xx TKNN number (Thouless-Kohmoto-Nightingale-den Nijs) xy 2 e h C 1 st Chern number 1 2 C d k k A kx, ky 2 i filled band A k, k k k x y k x, k k k y integer valued TKNN (1982); Kohmoto (1985) = number of edge modes crossing E F Berry connection bulk-edge correspondence
7 Systematic understanding of topological phases? Relationships to the symmetry and the dimensionality? A system of non-interacting fermion is classified into 10 Altland-Zirnbauer classes 5 classes of non-trivial TI/TSC for each dimension
8 Table of topological insulators/superconductors TRS PHS CS d=1 d=2 d=3 Standard (Wigner-Dyson) A (unitary) AI (orthogonal) AII (symplectic) Z QSHE IQHE -- Z 2 Z 2 Z 2 TPI Chiral AIII (chiral unitary) BDI (chiral orthogonal) Z -- Z polyacetylene (SSH) Z CII (chiral symplectic) Z -- Z 2 BdG D (p-wave SC) C (d-wave SC) DIII (p-wave TRS SC) p SC p+ip SC Z 2 Z -- d+id SC -- Z -- Z 2 Z 2 Z 3 He-B CI (d-wave TRS SC) Z Schnyder, Ryu, Furusaki, and Ludwig, PRB (2008)
9 Ten Altland-Zirnbauer symmetry classes Bilinear Hamiltonian: Fully block-diagonalized Hamiltonian matrix Three generic symmetries: -- Time-reversal symmetry Without B or magnetization -- Particle hole symmetry BdG equation, (superconductors) -- Chiral symmetry Sublattice symmetry, combination of TC
10 Derivation of the topological periodic table: Dirac Hamiltonian and topological phase Gapped phase (Assumption: Any gapped Hamiltonian can be deformed into the Dirac form) If Dirac mass g0 is unique, Massive Dirac Hamiltonian Topologically distinct phases m m<0 m=0 m>0
11 Derivation of the topological periodic table: Dirac Hamiltonian and topological phase Gapped phase (Assumption: Any gapped Hamiltonian can be deformed into the Dirac form) If Dirac mass g0 is unique, Massive Dirac Hamiltonian Topologically distinct phases Classification of topological phases Classification of Dirac mass term m m<0 m=0 m>0
12 Clifford algebras Complex Clifford algebra: n generators: Real Clifford algebra: p+q generators: 2 (p+q) dim real vector space spanned by bases of combinations of e i s
13 Clifford algebra of real symmetry classes Time-reversal symmetry T Particle-hole symmetry C T and C are antiunitary (i) J represents for i, T only (AI & AII): e 0 =Jg 0, e 1 =T, e 2 =TJ, e 3 =g 1,, e 2+d =g d (ii) C only (D & C): e 0 =g 0, e 1 =C, e 2 =CJ, e 3 =Jg 1,, e 2+d =Jg d (iii) T and C (BDI, DIII, CII & CI): e 0 =g 0, e 1 =C, e 2 =CJ, e 3 =TCJ, e 4 =Jg 1,, e 3+d =Jg d
14 Symmetry constraints for class D: e 0 =g 0, e 1 =C, e 2 =ic, e 3 =ig 1,, e 2+d =ig d Kinetic terms g i Symmetry operators Mass term g 0 Clifford algebra
15 Classification of topological insulators = Classification of Dirac mass term (i) We consider extension of Clifford algebra without g 0 into algebra with g 0 T, C, g d T, C, g d, g 0 (ii) All the possible extensions ( classifying space ) = space of g 0 R q-p R q-p and in disconnected parts Distinct topological phases (iii) Topological classification = Kitaev 09
16 Bott periodicity (i) (ii) (iii) Complex and real K-theory
17 Topological periodic table from Clifford algebra
18 Topological crystalline insulator Topological insulator with time-reversal + Reflection symmetry (Z 2 Z) -- SnTe compounds with even # of surface Dirac cones Experiments (SnTe/PbTe): Tanaka et al., Nat. Phys Xu et al., Nat. Commun Dziawa et al., Nat. Mat Hsieh et al. Nat. Commun. 2012
19 Reflection symmetry Reflection symmetry with 1-direction Dirac Hamiltonian x1 R ht, hc gives an additional chiral symmetry M anti-commutes with all g i s New generator of Clifford algebra
20 Topological periodic table with a reflection symmetry and R + R ++ R - R -- SnTe (class AII+R - d=3)
21 Interaction effects on topological insulators and superconductors
22 Breakdown of non-interacting topological phases labeled by Z with interactions Time-reversal symmetric Majorana chain (1D class BDI) Z Z 8 Time-reversal symmetric 3D topological SC (3D class DIII) Z Z 16 8 Majorana zero modes at the boundary can be gapped without breaking TRS. 16 Dirac surface fermions can be gapped without breaking TRS. Fidkowski and Kitaev, PRB (2010), PRB (2011) Kitaev (2011), Fidkowski etal. PRX (2013), Metlitski, Kane & Fisher. (2014),.
23 Aim: Systematic study of the breakdown of Z classification - Stability analysis of boundary gapless states against interactions in any dimension and any symmetry class Nonlinear sigma model - Applications to the tenfold way and other topological phases (topological crystalline insulators) Result:
24 Nonlinear sigma model approach ν copies of gapless boundary states ν copies Boundary massless Dirac fermions + quartic interactions β 1, β 2,, β N a, b: anti-commuting gamma matrices a respects symmetries, b can be odd under some symmetry operations. Hubbard-Stratonovich transformtation cf. You and Xu, PRB (2014), Kitaev s talk (2015) Dynamical Dirac masses
25 Integration of fermions Saddle point approximation + including fluctuations about the direction in which f freezes Nonlinear sigma model + topological term Abanov, Wiegmann Nucl. Phys. B (2000) Target space of NLSM is a sphere generated by N(n) anticommuting dynamical masses b s
26 Topological term in NLSM The presence or absence of a topological term is determined by the homotopy group of the target space. π 0 S N ν 1 0 domain wall Nontrivial homotopy group π 1 S N ν 1 0 π d S N ν 1 0 π d+1 S N ν 1 0 vortex Wess-Zumino term Topological term in NLSM Boundary states remain gapless Condition for the breakdown π D S N ν 1 = 0 for D = 0,, d + 1 Topological defects in the dynamical mass bind fermion zero-energy states. ν min : the minimum ν satisfying the above condition,
27 ν = 1 Example: 3D class DIII (3He-B phase) Bulk: Boundary: ν copies Dynamical mass: n n Real symmetric matrix Dynamical masses break T, but preserves C. Space of masses (2D class D) = real Grassmannians: R 0 = n=1: n=2: n=4: n=8:
28 Higher dimensions Z 2 entries are stable. Z in even dimensions is stable. Z in odd dimensions is unstable.
29 3D topological crystalline insulator (SnTe) (TRS+ reflection Z classification) Boundary: Hsieh et al BdG: R 0 = Minimal TCI = 2copies of 3He-B phase
30 Summary and outlook Classification of topological crystalline insulators General spatial symmetry that cannot fit into Clifford algebras? Twisted equivariant K-theory? T. Morimoto, A.Furusaki, Phys. Rev. B 88, (2013). Effects of interactions on topological insulators Nonlinear sigma model analysis over spherical target spaces More general interactions? New topological phases that emerges with interactions? T. Morimoto, A.Furusaki, C.Mudry, Phys. Rev. B 92, (2015)
Single particle Green s functions and interacting topological insulators
1 Single particle Green s functions and interacting topological insulators Victor Gurarie Nordita, Jan 2011 Topological insulators are free fermion systems characterized by topological invariants. 2 In
More informationModern Topics in Solid-State Theory: Topological insulators and superconductors
Modern Topics in Solid-State Theory: Topological insulators and superconductors Andreas P. Schnyder Max-Planck-Institut für Festkörperforschung, Stuttgart Universität Stuttgart January 2016 Lecture Four:
More informationCrystalline Symmetry and Topology. YITP, Kyoto University Masatoshi Sato
Crystalline Symmetry and Topology YITP, Kyoto University Masatoshi Sato In collaboration with Ken Shiozaki (YITP) Kiyonori Gomi (Shinshu University) Nobuyuki Okuma (YITP) Ai Yamakage (Nagoya University)
More informationA Short Introduction to Topological Superconductors
A Short Introduction to Topological Superconductors --- A Glimpse of Topological Phases of Matter Jyong-Hao Chen Condensed Matter Theory, PSI & Institute for Theoretical Physics, ETHZ Dec. 09, 2015 @ Superconductivity
More informationSymmetric Surfaces of Topological Superconductor
Symmetric Surfaces of Topological Superconductor Sharmistha Sahoo Zhao Zhang Jeffrey Teo Outline Introduction Brief description of time reversal symmetric topological superconductor. Coupled wire model
More informationTopological Insulators
Topological Insulators Aira Furusai (Condensed Matter Theory Lab.) = topological insulators (3d and 2d) Outline Introduction: band theory Example of topological insulators: integer quantum Hall effect
More informationTopological Phases of Matter Out of Equilibrium
Topological Phases of Matter Out of Equilibrium Nigel Cooper T.C.M. Group, Cavendish Laboratory, University of Cambridge Solvay Workshop on Quantum Simulation ULB, Brussels, 18 February 2019 Max McGinley
More informationOn the K-theory classification of topological states of matter
On the K-theory classification of topological states of matter (1,2) (1) Department of Mathematics Mathematical Sciences Institute (2) Department of Theoretical Physics Research School of Physics and Engineering
More informationC. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 1 / fermio 93. quartic interactions
The breakdown of the topological classification Z for gapped phases of noninteracting fermions by quartic interactions Christopher Mudry 1 Takahiro Morimoto 2,3 Akira Furusaki 2 1 Paul Scherrer Institut,
More informationDisordered topological insulators with time-reversal symmetry: Z 2 invariants
Keio Topo. Science (2016/11/18) Disordered topological insulators with time-reversal symmetry: Z 2 invariants Hosho Katsura Department of Physics, UTokyo Collaborators: Yutaka Akagi (UTokyo) Tohru Koma
More informationarxiv: v1 [cond-mat.mes-hall] 13 May 2009
Classification of Topological Insulators and Superconductors arxiv:0905.2029v1 [cond-mat.mes-hall] 13 May 2009 Andreas P. Schnyder, Shinsei Ryu, Akira Furusaki and Andreas W. W. Ludwig Kavli Institute
More informationClassification of Crystalline Topological Phases with Point Group Symmetries
Classification of Crystalline Topological Phases with Point Group Symmetries Eyal Cornfeld - Tel Aviv University Adam Chapman - Tel Hai Academic College Upper Galilee Bad Honnef Physics School on Gauge
More informationKonstantin Y. Bliokh, Daria Smirnova, Franco Nori. Center for Emergent Matter Science, RIKEN, Japan. Science 348, 1448 (2015)
Konstantin Y. Bliokh, Daria Smirnova, Franco Nori Center for Emergent Matter Science, RIKEN, Japan Science 348, 1448 (2015) QSHE and topological insulators The quantum spin Hall effect means the presence
More informationTopological invariants for 1-dimensional superconductors
Topological invariants for 1-dimensional superconductors Eddy Ardonne Jan Budich 1306.4459 1308.soon SPORE 13 2013-07-31 Intro: Transverse field Ising model H TFI = L 1 i=0 hσ z i + σ x i σ x i+1 σ s:
More informationReducing and increasing dimensionality of topological insulators
Reducing and increasing dimensionality of topological insulators Anton Akhmerov with Bernard van Heck, Cosma Fulga, Fabian Hassler, and Jonathan Edge PRB 85, 165409 (2012), PRB 89, 155424 (2014). ESI,
More informationK-theory in Condensed Matter Physics
(1,2) (1) Department of Mathematics Mathematical Sciences Institute (2) Department of Theoretical Physics Research School of Physics and Engineering The Australian National University Canberra, AUSTRALIA
More informationClassification of topological quantum matter with reflection symmetries
Classification of topological quantum matter with reflection symmetries Andreas P. Schnyder Max Planck Institute for Solid State Research, Stuttgart June 14th, 2016 SPICE Workshop on New Paradigms in Dirac-Weyl
More informationTopological nonsymmorphic crystalline superconductors
UIUC, 10/26/2015 Topological nonsymmorphic crystalline superconductors Chaoxing Liu Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA Chao-Xing Liu, Rui-Xing
More informationTopological Insulators and Superconductors
Topological Insulators and Superconductors Lecture #1: Topology and Band Theory Lecture #: Topological Insulators in and 3 dimensions Lecture #3: Topological Superconductors, Majorana Fermions an Topological
More informationTime Reversal Invariant Ζ 2 Topological Insulator
Time Reversal Invariant Ζ Topological Insulator D Bloch Hamiltonians subject to the T constraint 1 ( ) ΘH Θ = H( ) with Θ = 1 are classified by a Ζ topological invariant (ν =,1) Understand via Bul-Boundary
More informationSymmetry, Topology and Phases of Matter
Symmetry, Topology and Phases of Matter E E k=λ a k=λ b k=λ a k=λ b Topological Phases of Matter Many examples of topological band phenomena States adiabatically connected to independent electrons: - Quantum
More informationdisordered topological matter time line
disordered topological matter time line disordered topological matter time line 80s quantum Hall SSH quantum Hall effect (class A) quantum Hall effect (class A) 1998 Nobel prize press release quantum Hall
More informationIntroduction to topological insulators. Jennifer Cano
Introduction to topological insulators Jennifer Cano Adapted from Charlie Kane s Windsor Lectures: http://www.physics.upenn.edu/~kane/ Review article: Hasan & Kane Rev. Mod. Phys. 2010 What is an insulator?
More informationInterpolating between Wishart and inverse-wishart distributions
Interpolating between Wishart and inverse-wishart distributions Topological phase transitions in 1D multichannel disordered wires with a chiral symmetry Christophe Texier December 11, 2015 with Aurélien
More informationTopological Electromagnetic and Thermal Responses of Time-Reversal Invariant Superconductors and Chiral-Symmetric band insulators
Topological Electromagnetic and Thermal Responses of Time-Reversal Invariant Superconductors and Chiral-Symmetric band insulators Satoshi Fujimoto Dept. Phys., Kyoto University Collaborator: Ken Shiozaki
More informationEffective Field Theories of Topological Insulators
Effective Field Theories of Topological Insulators Eduardo Fradkin University of Illinois at Urbana-Champaign Workshop on Field Theoretic Computer Simulations for Particle Physics and Condensed Matter
More informationTopological Phases in Floquet Systems
Rahul Roy University of California, Los Angeles June 2, 2016 arxiv:1602.08089, arxiv:1603.06944 Post-doc: Fenner Harper Outline 1 Introduction 2 Free Fermionic Systems 3 Interacting Systems in Class D
More informationOrganizing Principles for Understanding Matter
Organizing Principles for Understanding Matter Symmetry Conceptual simplification Conservation laws Distinguish phases of matter by pattern of broken symmetries Topology Properties insensitive to smooth
More informationSymmetries in Quantum Transport : From Random Matrix Theory to Topological Insulators. Philippe Jacquod. U of Arizona
Symmetries in Quantum Transport : From Random Matrix Theory to Topological Insulators Philippe Jacquod U of Arizona UA Phys colloquium - feb 1, 2013 Continuous symmetries and conservation laws Noether
More informationUnderdoped superconducting cuprates as topological superconductors
Underdoped superconducting cuprates as topological superconductors Yuan-Ming Lu 1,2, Tao Xiang 3 & Dung-Hai Lee 1,2 SUPPLEMENTARY INFORMATION 1 Department of Physics, University of California, Berkeley,
More informationQuantitative Mappings from Symmetry to Topology
Z. Song, Z. Fang and CF, PRL 119, 246402 (2017) CF and L. Fu, arxiv:1709.01929 Z. Song, T. Zhang, Z. Fang and CF arxiv:1711.11049 Z. Song, T. Zhang and CF arxiv:1711.11050 Quantitative Mappings from Symmetry
More informationarxiv: v1 [math-ph] 31 Jan 2016
February 2, 2016 1:43 ws-procs961x669 WSPC Proceedings - 9.61in x 6.69in ProdanICMP2015 page 1 1 Topological Insulators at Strong Disorder Emil Prodan Physics Department, Yeshiva University, New York,
More informationAnderson localization, topology, and interaction
Anderson localization, topology, and interaction Pavel Ostrovsky in collaboration with I. V. Gornyi, E. J. König, A. D. Mirlin, and I. V. Protopopov PRL 105, 036803 (2010), PRB 85, 195130 (2012) Cambridge,
More informationQuantum Quenches in Chern Insulators
Quantum Quenches in Chern Insulators Nigel Cooper Cavendish Laboratory, University of Cambridge CUA Seminar M.I.T., November 10th, 2015 Marcello Caio & Joe Bhaseen (KCL), Stefan Baur (Cambridge) M.D. Caio,
More informationMany-body topological invariants for topological superconductors (and insulators)
Many-body topological invariants for topological superconductors (and insulators) Shinsei Ryu The University of Chicago June 6, 2017 Outline Motivations: the Kitaev chain with interactions The kitaev chain
More informationIntroductory lecture on topological insulators. Reza Asgari
Introductory lecture on topological insulators Reza Asgari Workshop on graphene and topological insulators, IPM. 19-20 Oct. 2011 Outlines -Introduction New phases of materials, Insulators -Theory quantum
More informationTopological protection, disorder, and interactions: Life and death at the surface of a topological superconductor
Topological protection, disorder, and interactions: Life and death at the surface of a topological superconductor Matthew S. Foster Rice University March 14 th, 2014 Collaborators: Emil Yuzbashyan (Rutgers),
More informationField Theory Description of Topological States of Matter. Andrea Cappelli INFN, Florence (w. E. Randellini, J. Sisti)
Field Theory Description of Topological States of Matter Andrea Cappelli INFN, Florence (w. E. Randellini, J. Sisti) Topological States of Matter System with bulk gap but non-trivial at energies below
More informationTwisted Equivariant Matter
Twisted Equivariant Matter Gregory Moore, Rutgers University, SCGP, June 12, 2013 References: 1. D. Freed and G. Moore, Twisted Equivariant Matter, arxiv:1208.5055 2. G. Moore, Quantum Symmetries and K
More informationFloquet theory of photo-induced topological phase transitions: Application to graphene
Floquet theory of photo-induced topological phase transitions: Application to graphene Takashi Oka (University of Tokyo) T. Kitagawa (Harvard) L. Fu (Harvard) E. Demler (Harvard) A. Brataas (Norweigian
More informationClassification of Symmetry Protected Topological Phases in Interacting Systems
Classification of Symmetry Protected Topological Phases in Interacting Systems Zhengcheng Gu (PI) Collaborators: Prof. Xiao-Gang ang Wen (PI/ PI/MIT) Prof. M. Levin (U. of Chicago) Dr. Xie Chen(UC Berkeley)
More informationField Theory Description of Topological States of Matter
Field Theory Description of Topological States of Matter Andrea Cappelli, INFN Florence (w. E. Randellini, J. Sisti) Outline Topological states of matter Quantum Hall effect: bulk and edge Effective field
More informationTopological Insulators in 3D and Bosonization
Topological Insulators in 3D and Bosonization Andrea Cappelli, INFN Florence (w. E. Randellini, J. Sisti) Outline Topological states of matter: bulk and edge Fermions and bosons on the (1+1)-dimensional
More informationarxiv: v2 [cond-mat.str-el] 22 Oct 2018
Pseudo topological insulators C. Yuce Department of Physics, Anadolu University, Turkey Department of Physics, Eskisehir Technical University, Turkey (Dated: October 23, 2018) arxiv:1808.07862v2 [cond-mat.str-el]
More informationarxiv: v2 [cond-mat.mes-hall] 31 Mar 2016
Journal of the Physical Society of Japan LETTERS Entanglement Chern Number of the ane Mele Model with Ferromagnetism Hiromu Araki, Toshikaze ariyado,, Takahiro Fukui 3, and Yasuhiro Hatsugai, Graduate
More informationCenke Xu. Quantum Phase Transitions between Bosonic Symmetry Protected Topological States without sign problem 许岑珂
Quantum Phase Transitions between Bosonic Symmetry Protected Topological States without sign problem Cenke Xu 许岑珂 University of California, Santa Barbara Quantum Phase Transitions between bosonic Symmetry
More information5 Topological insulator with time-reversal symmetry
Phys62.nb 63 5 Topological insulator with time-reversal symmetry It is impossible to have quantum Hall effect without breaking the time-reversal symmetry. xy xy. If we want xy to be invariant under, xy
More informationSurface Majorana Fermions in Topological Superconductors. ISSP, Univ. of Tokyo. Nagoya University Masatoshi Sato
Surface Majorana Fermions in Topological Superconductors ISSP, Univ. of Tokyo Nagoya University Masatoshi Sato Kyoto Tokyo Nagoya In collaboration with Satoshi Fujimoto (Kyoto University) Yoshiro Takahashi
More informationTopological insulators. Pavel Buividovich (Regensburg)
Topological insulators Pavel Buividovich (Regensburg) Hall effect Classical treatment Dissipative motion for point-like particles (Drude theory) Steady motion Classical Hall effect Cyclotron frequency
More informationMany-body topological invariants for topological superconductors (and insulators)
Many-body topological invariants for topological superconductors (and insulators) Shinsei Ryu The University of Chicago July 5, 2017 Outline Motivations: the Kitaev chain with interactions The kitaev chain
More informationComposite Dirac liquids
Composite Dirac liquids Composite Fermi liquid non-interacting 3D TI surface Interactions Composite Dirac liquid ~ Jason Alicea, Caltech David Mross, Andrew Essin, & JA, Physical Review X 5, 011011 (2015)
More informationFermionic partial transpose and non-local order parameters for SPT phases of fermions
Fermionic partial transpose and non-local order parameters for SPT phases of fermions Ken Shiozaki RIKEN Corroborators: Hassan Shapourian Shinsei Ryu Kiyonori Gomi University of Chicago University of Chicago
More informationTopological Phases in One Dimension
Topological Phases in One Dimension Lukasz Fidkowski and Alexei Kitaev arxiv:1008.4138 Topological phases in 2 dimensions: - Integer quantum Hall effect - quantized σ xy - robust chiral edge modes - Fractional
More informationKITP miniprogram, Dec. 11, 2008
1. Magnetoelectric polarizability in 3D insulators and experiments! 2. Topological insulators with interactions (3. Critical Majorana fermion chain at the QSH edge) KITP miniprogram, Dec. 11, 2008 Joel
More informationFractional Abelian topological phases of matter for fermions in two-dimensional space
Fractional Abelian topological phases of matter for fermions in two-dimensional space Christopher Mudry Condensed Matter Theory Group, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland 1 Introduction
More informationTopological Defects inside a Topological Band Insulator
Topological Defects inside a Topological Band Insulator Ashvin Vishwanath UC Berkeley Refs: Ran, Zhang A.V., Nature Physics 5, 289 (2009). Hosur, Ryu, AV arxiv: 0908.2691 Part 1: Outline A toy model of
More informationarxiv: v2 [hep-th] 1 Aug 2010
IPMU10-0126 Topological Insulators and Superconductors from String Theory Shinsei Ryu Department of Physics, University of California, Berkeley, CA 94720, USA Tadashi Takayanagi Institute for the Physics
More informationDefects in topologically ordered states. Xiao-Liang Qi Stanford University Mag Lab, Tallahassee, 01/09/2014
Defects in topologically ordered states Xiao-Liang Qi Stanford University Mag Lab, Tallahassee, 01/09/2014 References Maissam Barkeshli & XLQ, PRX, 2, 031013 (2012) Maissam Barkeshli, Chaoming Jian, XLQ,
More informationSymmetry Protected Topological Insulators and Semimetals
Symmetry Protected Topological Insulators and Semimetals I. Introduction : Many examples of topological band phenomena II. Recent developments : - Line node semimetal Kim, Wieder, Kane, Rappe, PRL 115,
More informationTopological insulator with time-reversal symmetry
Phys620.nb 101 7 Topological insulator with time-reversal symmetry Q: Can we get a topological insulator that preserves the time-reversal symmetry? A: Yes, with the help of the spin degree of freedom.
More informationFloquet Topological Insulators and Majorana Modes
Floquet Topological Insulators and Majorana Modes Manisha Thakurathi Journal Club Centre for High Energy Physics IISc Bangalore January 17, 2013 References Floquet Topological Insulators by J. Cayssol
More informationQuantum transport of 2D Dirac fermions: the case for a topological metal
Quantum transport of 2D Dirac fermions: the case for a topological metal Christopher Mudry 1 Shinsei Ryu 2 Akira Furusaki 3 Hideaki Obuse 3,4 1 Paul Scherrer Institut, Switzerland 2 University of California
More informationQuantum interference meets topology: Quantum Hall effect, topological insulators, and graphene. Alexander D. Mirlin
Quantum interference meets topology: Quantum Hall effect, topological insulators, and graphene Alexander D. Mirlin Karlsruhe Institute of Technology & PNPI St. Petersburg P. Ostrovsky, Karlsruhe Institute
More informationTopological classification of 1D symmetric quantum walks
Topological classification of 1D symmetric quantum walks Albert H. Werner QMath University of Copenhagen Together with: C. Cedzic, T. Geib, F. A. Grünbaum, C. Stahl, L. Velazques, R. F. Werner Phase classification
More informationIntroduction to topological insulators
Introduction to topological insulators Janos Asboth1, Laszlo Oroszlany2, Andras Palyi3 1: Wigner Research Centre for Physics, Hungarian Academy of Sciences 2: Eotvos University, Budapest 3: Technical University,
More informationTopological Kondo Insulator SmB 6. Tetsuya Takimoto
Topological Kondo Insulator SmB 6 J. Phys. Soc. Jpn. 80 123720, (2011). Tetsuya Takimoto Department of Physics, Hanyang University Collaborator: Ki-Hoon Lee (POSTECH) Content 1. Introduction of SmB 6 in-gap
More informationTopological states of matter in correlated electron systems
Seminar @ Tsinghua, Dec.5/2012 Topological states of matter in correlated electron systems Qiang-Hua Wang National Lab of Solid State Microstructures, Nanjing University, Nanjing 210093, China Collaborators:Dunghai
More informationSymmetry Protected Topological Phases of Matter
Symmetry Protected Topological Phases of Matter T. Senthil (MIT) Review: T. Senthil, Annual Reviews of Condensed Matter Physics, 2015 Topological insulators 1.0 Free electron band theory: distinct insulating
More informationarxiv: v1 [cond-mat.other] 20 Apr 2010
Characterization of 3d topological insulators by 2d invariants Rahul Roy Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford, OX1 3NP, UK arxiv:1004.3507v1 [cond-mat.other] 20 Apr 2010
More informationarxiv: v2 [cond-mat.str-el] 16 Aug 2016
Periodic able for Floquet opological Insulators Rahul Roy and Fenner Harper Department of Physics and Astronomy, University of California, Los Angeles, California USA Dated: August 18, 16 arxiv:163.6944v
More informationExploring topological states with cold atoms and photons
Exploring topological states with cold atoms and photons Theory: Takuya Kitagawa, Dima Abanin, Erez Berg, Mark Rudner, Liang Fu, Takashi Oka, Immanuel Bloch, Eugene Demler Experiments: I. Bloch s group
More informationTopological properties of superconducting chains
Bachelor's thesis Theoretical Physics Topological properties of superconducting chains Kim Pöyhönen 213 Advisor: Supervisor: Teemu Ojanen Tommy Ahlgren Helsinki University Department of Physics Postbox
More informationAntiferromagnetic topological insulators
Antiferromagnetic topological insulators Roger S. K. Mong, Andrew M. Essin, and Joel E. Moore, Department of Physics, University of California, Berkeley, California 9470, USA Materials Sciences Division,
More informationSPT: a window into highly entangled phases
SPT: a window into highly entangled phases T. Senthil (MIT) Collaborators: Chong Wang, A. Potter Why study SPT? 1. Because it may be there... Focus on electronic systems with realistic symmetries in d
More informationTopological delocalization of two-dimensional massless fermions
- CMP Meets HEP at IPMU Kashiwa /10/010 - Topoloical delocalization of two-dimensional massless fermions Kentaro Nomura (Tohoku University) collaborators Shinsei Ryu (Berkeley) Mikito Koshino (Titech)
More informationThe space group classification of topological band insulators
arxiv:1209.2610v2 [cond-mat.mes-hall] 19 Nov 2012 The space group classification of topological band insulators Robert-Jan Slager 1, Andrej Mesaros 2, Vladimir Juričić 1 and Jan Zaanen 1 1 Instituut-Lorentz
More informationTopology of electronic bands and Topological Order
Topology of electronic bands and Topological Order R. Shankar The Institute of Mathematical Sciences, Chennai TIFR, 26 th April, 2011 Outline IQHE and the Chern Invariant Topological insulators and the
More informationBraid Group, Gauge Invariance and Topological Order
Braid Group, Gauge Invariance and Topological Order Yong-Shi Wu Department of Physics University of Utah Topological Quantum Computing IPAM, UCLA; March 2, 2007 Outline Motivation: Topological Matter (Phases)
More informationKouki Nakata. University of Basel. KN, S. K. Kim (UCLA), J. Klinovaja, D. Loss (2017) arxiv:
Magnon Transport Both in Ferromagnetic and Antiferromagnetic Insulating Magnets Kouki Nakata University of Basel KN, S. K. Kim (UCLA), J. Klinovaja, D. Loss (2017) arxiv:1707.07427 See also review article
More informationConstructing Landau Formalism for Topological Order: Spin Chains and Ladders
Constructing Landau Formalism for Topological Order: Spin Chains and Ladders Gennady Y. Chitov Laurentian University Sudbury, Canada Talk at Washington University in St. Louis, October 20, 2016 Collaborators:
More informationIs the composite fermion a Dirac particle?
Is the composite fermion a Dirac particle? Dam T. Son GGI conference Gauge/gravity duality 2015 Ref.: 1502.03446 Plan Plan Fractional quantum Hall effect Plan Fractional quantum Hall effect Composite fermion
More informationEmergent topological phenomena in antiferromagnets with noncoplanar spins
Emergent topological phenomena in antiferromagnets with noncoplanar spins - Surface quantum Hall effect - Dimensional crossover Bohm-Jung Yang (RIKEN, Center for Emergent Matter Science (CEMS), Japan)
More informationTopological Insulators
Topological Insulators A new state of matter with three dimensional topological electronic order L. Andrew Wray Lawrence Berkeley National Lab Princeton University Surface States (Topological Order in
More informationInteractions in Topological Matter
Interactions in Topological Matter Christopher Mudry 1 1 Paul Scherrer Institute, Switzerland Harish-Chandra Research Institute, Allahabad, February 9-20 2015 C. Mudry (PSI) Interactions in Topological
More informationarxiv: v3 [cond-mat.supr-con] 4 Apr 2017
Topological superconductors: a review Masatoshi Sato 1, and Yoichi Ando 2, 1 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan 2 Physics Institute II, University of Cologne,
More informationarxiv: v2 [cond-mat.mes-hall] 29 Oct 2013
Topological invariant for generic 1D time reversal symmetric superconductors in class DIII Jan Carl Budich, Eddy Ardonne Department of Physics, Stockholm University, SE-106 91 Stockholm, Sweden Dated:
More informationSSH Model. Alessandro David. November 3, 2016
SSH Model Alessandro David November 3, 2016 Adapted from Lecture Notes at: https://arxiv.org/abs/1509.02295 and from article: Nature Physics 9, 795 (2013) Motivations SSH = Su-Schrieffer-Heeger Polyacetylene
More informationQuantum Spin Liquids and Majorana Metals
Quantum Spin Liquids and Majorana Metals Maria Hermanns University of Cologne M.H., S. Trebst, PRB 89, 235102 (2014) M.H., K. O Brien, S. Trebst, PRL 114, 157202 (2015) M.H., S. Trebst, A. Rosch, arxiv:1506.01379
More informationWiring Topological Phases
1 Wiring Topological Phases Quantum Condensed Matter Journal Club Adhip Agarwala Department of Physics Indian Institute of Science adhip@physics.iisc.ernet.in February 4, 2016 So you are interested in
More information(1) Topological terms and metallic transport (2) Dynamics as a probe of Majorana fermions
(1) Topological terms and metallic transport (2) Dynamics as a probe of Majorana fermions Harvard, September 16, 2014 Joel Moore University of California, Berkeley, and Lawrence Berkeley National Laboratory
More informationTopological Superconductivity and Superfluidity
Topological Superconductivity and Superfluidity SLAC-PUB-13926 Xiao-Liang Qi, Taylor L. Hughes, Srinivas Raghu and Shou-Cheng Zhang Department of Physics, McCullough Building, Stanford University, Stanford,
More informationDirac-Fermion-Induced Parity Mixing in Superconducting Topological Insulators. Nagoya University Masatoshi Sato
Dirac-Fermion-Induced Parity Mixing in Superconducting Topological Insulators Nagoya University Masatoshi Sato In collaboration with Yukio Tanaka (Nagoya University) Keiji Yada (Nagoya University) Ai Yamakage
More informationElectron transport and quantum criticality in disordered graphene. Alexander D. Mirlin
Electron transport and quantum criticality in disordered graphene Alexander D. Mirlin Research Center Karslruhe & University Karlsruhe & PNPI St. Petersburg P. Ostrovsky, Research Center Karlsruhe & Landau
More informationTopological insulator (TI)
Topological insulator (TI) Haldane model: QHE without Landau level Quantized spin Hall effect: 2D topological insulators: Kane-Mele model for graphene HgTe quantum well InAs/GaSb quantum well 3D topological
More informationNotes on Topological Insulators and Quantum Spin Hall Effect. Jouko Nieminen Tampere University of Technology.
Notes on Topological Insulators and Quantum Spin Hall Effect Jouko Nieminen Tampere University of Technology. Not so much discussed concept in this session: topology. In math, topology discards small details
More informationTopological order in the pseudogap metal
HARVARD Topological order in the pseudogap metal High Temperature Superconductivity Unifying Themes in Diverse Materials 2018 Aspen Winter Conference Aspen Center for Physics Subir Sachdev January 16,
More informationFermionic partial transpose fermionic entanglement and fermionic SPT phases
Fermionic partial transpose fermionic entanglement and fermionic SPT phases Shinsei Ryu University of Chicago November 7, 2017 Outline 1. Bosonic case (Haldane chain) What is partial tranpose? Why it is
More informationAnomalies and SPT phases
Anomalies and SPT phases Kazuya Yonekura, Kavli IPMU Based on A review of [1508.04715] by Witten [1607.01873] KY [1609.?????] Yuji Tachikawa and KY Introduction One of the motivations: What is the most
More informationLecture notes on topological insulators
Lecture notes on topological insulators Ming-Che Chang Department of Physics, National Taiwan Normal University, Taipei, Taiwan Dated: May 8, 07 I. D p-wave SUPERCONDUCTOR Here we study p-wave SC in D
More informationExotic Phenomena in Topological Insulators and Superconductors
SPICE Workshop on Spin Dynamics in the Dirac System Schloss Waldthausen, Mainz, 6 June 2017 Exotic Phenomena in Topological Insulators and Superconductors Yoichi Ando Physics Institute II, University of
More information